#8127272

order	512
nilpotency class	2
automorphism	?
homology	Abelianization (H₁) = C₂⁶ · Schur multiplier (H₂) = C₂¹²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, commutator, Frattini, socle, cosocle	C₂³	C₂⁶
prime radical	#41404	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4388 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 6 generators on 40 points:

(2,5)(6,8)(13,21)(15,22)(18,24)(20,23)(25,26)(27,36)(28,38)(29,32)(30,40)(31,35)(33,39)(34,37);(3,7)(6,8)(9,10,14,16)(11,17,19,12)(13,15,21,22)(18,23,24,20)(25,27,31,29)(26,32,35,36)(28,37,39,30)(33,34,38,40);(1,2)(3,6)(4,5)(7,8)(9,11,14,19)(10,12,16,17)(13,18,21,24)(15,20,22,23)(25,26)(27,32)(28,38)(29,36)(30,34)(31,35)(33,39)(37,40);(1,3,4,7)(2,6,5,8)(9,12)(10,11)(13,20)(14,17)(15,18)(16,19)(21,23)(22,24)(25,28,31,39)(26,33,35,38)(27,37,29,30)(32,34,36,40);(1,2)(3,6)(4,5)(7,8)(9,12)(10,11)(13,20)(14,17)(15,18)(16,19)(21,23)(22,24)(25,29,31,27)(26,32,35,36)(28,30,39,37)(33,34,38,40);(9,13)(10,15)(11,18)(12,20)(14,21)(16,22)(17,23)(19,24)(25,30)(26,34)(27,28)(29,39)(31,37)(32,38)(33,36)(35,40)

Polycyclic presentation:

2^9|b^2=gi,c^2=i,d^2=gh,e^2=g,b^a=bg,c^a=ch,d^a=dg,e^a=eh,f^a=fgi,c^b=ci,d^b=dh,d^c=dgi,e^c=egi

Conjugacy classes & character table

86 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8034468);